by [[1]}, Proposition 7.2] and Proposition REF -(ii), the perfect relative obstruction theory for \(\mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho \)  over \(\mathfrak {C}_{0}^{\dagger ,\varrho }\)  can be also defined by pulling back of \(\Phi : \mathbf {E}^\bullet \rightarrow \mathbb {L}_{\mathfrak {PT}^P_{\mathfrak {X}/\mathfrak {C}}/\mathfrak {C}^P}\) , and \(\zeta _{\varrho }^![\mathfrak {PT}_{\mathfrak {X}/\mathfrak {C}}^P]^\mathrm {vir}=[\mathfrak {PT}_{\mathfrak {X}_{0}^\dagger /\mathfrak {C}_{0}^\dagger }^\varrho ]^\mathrm {vir}\) .
